Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2608.10390

A fifth-order divergence-free finite difference Hermite WENO scheme for ideal magnetohydrodynamics

Peiwen Chen·Shi Jin·Jianxian Qiu·Zhuang Zhao
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we present a fifth-order finite difference divergence-free Hermite weighted essentially non-oscillatory (HWENO) scheme for the ideal magnetohydrodynamics (MHD) equations. In this framework, both the solution and its spatial partial derivatives are evolved in time and jointly employed in the spatial reconstruction procedure. A major challenge in MHD simulations is preserving the divergence-free constraint of the magnetic field, which is generally violated by standard numerical methods designed solely for hyperbolic conservation laws. To address this issue, we first solve the MHD equations within the HWENO framework for hyperbolic conservation laws, yielding a magnetic field divergence that remains zero up to high-order accuracy in smooth regions. Subsequently, we apply a correction that evenly distributes the divergence error among the partial derivatives involved in the divergence-free constraint, thereby rendering the magnetic field discretely divergence-free at the new time level. This approach offers several advantages. First, the scheme retains the conservation property, as only the partial derivatives of the numerical solution are corrected, leaving the conserved variables unchanged. Second, the correction applied to the partial derivatives of the magnetic field components introduces only a high-order perturbation, thereby preserving the overall accuracy. Third, the divergence-free treatment significantly enhances robustness, as most benchmark test cases cannot be run stably without such a correction. Fourth, the correction is a simple linear operation applied at each stage of the time integration, incurring negligible additional computational cost. Extensive numerical experiments demonstrate the accuracy, resolution, efficiency, effectiveness, and robustness of the proposed scheme.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Peiwen Chen, Shi Jin, Jianxian Qiu, Zhuang Zhao. 2026-08-11. A fifth-order divergence-free finite difference Hermite WENO scheme for ideal magnetohydrodynamics. https://arxiv.org/abs/2608.10390

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

How many continuous measurements are needed to learn a vector?

One can recover vectors from $\mathbb{R}^m$ with arbitrary precision, using only $\lceil \log_2(m)\rceil +1$ continuous measurements that are chosen adaptively. This surprising result is explained and discussed, and we present applications to infinite-dimensional approximation problems.

math.NA↗
arXiv2026

IterativeCUR: Large Rank-Adaptive Approximation From a Small Recycled Sketch

The computation of accurate low-rank matrix approximations is central to improving the scalability of various techniques in machine learning, uncertainty quantification, and control. Traditionally, low-rank approximations are constructed using SVD-based approaches such as truncated SVD or Randomized SVD. Although these SVD approaches---especially Randomized SVD---have proven to be very computationally efficient, other low-rank approximation methods can offer even greater performance. One such approach is the CUR decomposition, which forms a low-rank approximation using direct row and column subsets of a matrix. Because CUR uses direct matrix subsets, it is also often better able to preserve native matrix structures like sparsity or non-negativity than SVD-based approaches and can facilitate data interpretation in many contexts. This paper introduces IterativeCUR, which draws on previous work in randomized numerical linear algebra to build a new algorithm that is highly competitive compared to prior work. IterativeCUR is adaptive in the sense that it takes as an input parameter the desired tolerance $ε$ and outputs (with arbitrarily high probability) an approximation of error bounded by $ε$, rather than requiring an a priori guess of the numerical rank. IterativeCUR typically runs significantly faster than both existing CUR algorithms and techniques such as Randomized SVD. Its asymptotic complexity is $\mathcal{O}(mn + (m+n)r^2)$ for an $m\times n$ matrix of output rank $r$. IterativeCUR relies on a single small sketch from the matrix that is successively downdated as the algorithm proceeds. We demonstrate through extensive experiments that IterativeCUR achieves up to $4\times$ speed-up over state-of-the-art pivoting-on-sketch approaches with no loss of accuracy, and up to $40\times$ speed-up over rank-adaptive randomized SVD approaches.

math.NA↗
arXiv2026

Multigrid with Linear Storage Complexity

As the discretization error for the solution of a partial differential equation (PDE) decreases, the precision required to store the corresponding coefficients naturally increases. Storing the solution's finite element coefficients explicitly requires $\mathcal O(n \log n)$ bits of storage, where $n$ is the number of degrees of freedom (DoFs). This paper presents a full multigrid method to compute the solution in a compressed format that reduces the storage complexity of the solution and intermediate vectors to $\mathcal O(n)$ bits. This reduction allows a matrix-free implementation to solve elliptic PDEs with an overall linear space complexity. For problems limited by the memory capacity of current supercomputers, we expect a memory footprint reduction of about an order of magnitude compared to state-of-the-art mixed-precision methods. We demonstrate the applicability of our algorithm by solving two model problems. Depending on the PDE and polynomial degree, but irrespective of the problem size, the solution vector on the finest grid requires between 4 and 12 bits per DoF, and the residual and correction require 3 to 6 bits each. Additional data is stored on the coarse grids with modestly increasing bit widths toward coarser grids.

math.NA↗